I explain the shutdown problem: the problem of designing artificial agents that (1) shut down when a shutdown button is pressed, (2) don’t try to prevent or cause the pressing of the shutdown button, and (3) otherwise pursue goals competently. I prove three theorems that make the difficulty precise. These theorems show that agents satisfying some innocuous-seeming conditions will often try to prevent or cause the pressing of the shutdown button, even in cases where it’s costly to do so. And patience trades off against shutdownability: the more patient an agent, the greater the costs that agent is willing to incur to manipulate the shutdown button. I end by noting that these theorems can guide our search for solutions.